| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s − 4i·7-s + i·8-s − 9-s + i·12-s − 6i·13-s − 4·14-s + 16-s − 2i·17-s + i·18-s − 19-s − 4·21-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s − 1.51i·7-s + 0.353i·8-s − 0.333·9-s + 0.288i·12-s − 1.66i·13-s − 1.06·14-s + 0.250·16-s − 0.485i·17-s + 0.235i·18-s − 0.229·19-s − 0.872·21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9671607078\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9671607078\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 23 | \( 1 - 8iT - 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 + 10iT - 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 + 4iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.153685694424773513054847193544, −7.40976707091801464701384535056, −7.16643917689858276515281674832, −5.73685866075118849776137662862, −5.26851736532864244408542620485, −3.93355982653213919556634538434, −3.51319819809025100368643252288, −2.36823222183087952222672230080, −1.19556409811313956677762926692, −0.32847670179714550338281588766,
1.84806203903419605008932962558, 2.81561454243333712503913721757, 4.03559650739469845883376899103, 4.68713193681521560981260189493, 5.50364735398736847891208494219, 6.29011011853380053769828032362, 6.75990377403134833462161881900, 7.988183746019224980298770398170, 8.618374181995704999307424050565, 9.185434879203501535050741887237
